Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.

To show that a subset $W$ of a vector space $V$ is a subspace, we need to check that

the zero vector in $V$ is in $W$

for any two vectors $u,v \in W$, we have $u+v \in W$

for any scalar $c$ and any vector $u \in W$, we have $cu \in W$.

Solution.

(a) Is $S=\{f(x) \in V \mid f(0)=f(1)\}$ a subspace?

We show that $S$ is a subspace of the vector space $V$ by checking conditions (1)-(3) given in the hint above.
First note that the zero vector in $V$ is the zero function $\theta(x)$, that is, $\theta(x)=0$ for any $x \in [0,1]$.
Since we have $\theta(0)=0=\theta(1)$, the zero function $\theta(x)\in S$.
Condition (1) is met.

Now, take any $f(x), g(x) \in S$. By the defining relation of $S$, we have
\[f(0)=f(1), \quad g(0)=g(1).\]
Consider the addition $(f+g)(x)$. We have
\[(f+g)(0)=f(0)+g(0)=f(1)+g(1)=(f+g)(1)\]
and it follows that $(f+g)(x) \in S$.
Thus $S$ satisfies condition (2).

To check the condition (3), take any scalar $c \in \R$ and $f(x) \in S$.
Since $f(x)\in S$, we have $f(0)=f(1)$. The scalar multiplication $(cf)(x)$ satisfies
\[(cf)(0)=c\cdot f(0)=c\cdot f(1)= (cf)(0).\]
Thus $(cf)(x) \in S$.

Therefore, the subset $S$ satisfies conditions (1)-(3). Hence $S$ is a subspace of $V$.

(b) Is $T=\{f(x) \in V \mid f(0)=f(1)+3\}$ a subspace?

We claim that $T$ is not a subspace of the vector space $V$.
For example, the subset $T$ does not satisfy condition (1).

The zero vector of $V$ is the zero function $\theta(x)$.
Then we have
\[\theta(0)=0 \neq 0+3=\theta(1)+3,\]
and hence the zero vector $\theta(x) \in V$ is not in $W$.

Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\]
be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Proof. […]

Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis
Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
Here $p'(x)$ is the first derivative of $p(x)$ and […]

Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$
Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \]
where $\Z$ is the set of all integers.
[…]

Determine the Values of $a$ so that $W_a$ is a Subspace
For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?
Solution.
The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]

The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
Proof.
To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]

12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

Is the Set of All Orthogonal Matrices a Vector Space?
An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.
Consider the subset
\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\]
Prove or disprove that $W$ is a subspace of […]